Set-theoretical basis for real numbers

1950 ◽  
Vol 15 (4) ◽  
pp. 241-247
Author(s):  
Hao Wang
Keyword(s):  
Upper Bound ◽  
Theoretical Basis ◽  
The Other ◽  
Real Numbers ◽  
Full Generality ◽  
Ordered Field ◽  
Other Hand

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.

Non-homogeneous binary cubic forms

1951 ◽  
Vol 47 (3) ◽  
pp. 457-460 ◽  
Author(s):  
R. P. Bambah
Keyword(s):  
Lower Bound ◽  
Finite Volume ◽  
Upper Bound ◽  
The Other ◽  
Real Numbers ◽  
Cubic Form ◽  
Homogeneous Form ◽  
Other Hand ◽  
Image Position ◽  
Cubic Forms

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

Reënforcement of Rubber by Fillers

1929 ◽  
Vol 2 (1) ◽  
pp. 21-31
Author(s):  
John T. Blake
Keyword(s):  
Fatty Acids ◽  
Theoretical Basis ◽  
Polar Compounds ◽  
The Other ◽  
Rubber Matrix ◽  
Double Integral ◽  
Other Hand ◽  
Sound Theoretical Basis ◽  
Volume Range

Abstract Wiegand's “Δ A” function, as represented by the double integral of tensile with respect to elongation and volume of a pigment over the base mix, is a practical means of expressing the reënforcing effect of the filler. On the other hand, the volume range over which the bonding of the filler is greater than the strength of the rubber matrix gives a better understanding of the condition of the filler in the rubber. Apparently, the importance of fatty acids and other polar compounds in dispersions can be put on a sound theoretical basis.

ON SIMULATING A CLASS OF PARALLEL ARCHITECTURES

2006 ◽  
Vol 17 (01) ◽  
pp. 91-110 ◽  
Author(s):  
RODICA CETERCHI ◽  
MARIO J. PÉREZ-JIMÉNEZ
Keyword(s):  
Parallel Architectures ◽  
The Other ◽  
P System ◽  
Full Generality ◽  
Other Hand ◽  
Communication Graphs

The purpose of this paper is twofold. On one hand, we introduce the concept of P system with dynamic communication graphs in its full generality, independent of applications. On the other hand, we illustrate one application of it to the simulation of a class of parallel architectures. In this last direction we extend previous work concerned with the simulation of particular architectures.

scholarly journals Propuesta de metodología para el cálculo de los parámetros fundamentales del sistema de limpieza de las cosechadoras de cereales

2017 ◽  
Vol 4 (1) ◽  
pp. 141-145
Author(s):  
Idalberto Macías Socarrás ◽  
Benjamín Gaskin Espinosa ◽  
Antonio Barrera Amat ◽  
Lenni Ramírez Flores ◽  
Mercedes Arzube Mayorga
Keyword(s):  
Theoretical Basis ◽  
The Other ◽  
Agricultural Machinery ◽  
Cleaning System ◽  
Agricultural Mechanization ◽  
Other Hand ◽  
Agricultural Sciences ◽  
The Subject ◽  
Important System

El presente trabajo recoge, de forma resumida, una propuesta de metodología para el cálculo de los principales parámetros del sistema de limpieza de las cosechadoras de cereales; la misma para su mejor comprensión se divide en tres partes fundamentales: parámetros del sacudidor de paja, parámetros de la superficie de limpieza o tamiz y parámetros principales del ventilador. La fundamentación teórica de esta metodología, ayudará a una mejor comprensión del funcionamiento de este importante sistema para estudiantes y especialistas del tema. Consecuentemente,  constituye una herramienta de trabajo para los estudiantes de las carreras de las ciencias técnicas agropecuarias, para solución de tareas relacionadas con la mecanización agropecuaria y teoría de máquinas agrícolas. AbstractThe present paper summarizes a proposal for a methodology for the calculation of the main parameters of the cleaning system of grain harvesters; the same for its better understanding is divided in three fundamental parts: parameters of the straw shaker, parameters of the cleaning surface or sieve and main parameters of the fan. The theoretical basis of this methodology will help a better understanding of the functioning of this important system for students and specialists in the subject. On the other hand, it is a working tool for the students of agricultural sciences, to solve tasks related to agricultural mechanization and agricultural machinery theory.  

scholarly journals The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

Some James numbers of Stiefel manifolds

1982 ◽  
Vol 92 (1) ◽  
pp. 139-161 ◽  
Author(s):  
Hideaki Ōshima
Keyword(s):  
The Other ◽  
Stiefel Manifold ◽  
Complex Numbers ◽  
Real Numbers ◽  
The Real ◽  
Stiefel Manifolds ◽  
Other Hand ◽  
Usual Norm

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

Quadratic forms ‘à la’ local theory

1967 ◽  
Vol 63 (3) ◽  
pp. 579-586 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Quadratic Forms ◽  
The Other ◽  
Local Theory ◽  
Real Numbers ◽  
The Real ◽  
Other Hand ◽  
Non Local ◽  
Characteristic 2 ◽  
The One ◽  
Adic Case

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

scholarly journals Triebel-Lizorkin capacity and hausdorff measure in metric spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 617-624
Author(s):  
Nijjwal Karak
Keyword(s):  
Hausdorff Measure ◽  
Upper Bound ◽  
Metric Spaces ◽  
Measure Zero ◽  
Gauge Function ◽  
The Other ◽  
Other Hand ◽  
Zero Capacity

AbstractWe provide a upper bound for Triebel-Lizorkin capacity in metric settings in terms of Hausdorff measure. On the other hand, we also prove that the sets with zero capacity have generalized Hausdorff h-measure zero for a suitable gauge function h.

Analysis of Central Bursting Defects in Plane Strain Drawing and Extrusion

1986 ◽  
Vol 108 (4) ◽  
pp. 317-321 ◽  
Author(s):  
B. Avitzur ◽  
J. C. Choi
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
The Other ◽  
Major Conclusion ◽  
Upper Bound Theorem ◽  
Other Hand ◽  
Identical Manner ◽  
Bound Theorem ◽  
Inclined Angle ◽  
Proportional Flow

Based on the upper-bound theorem in limit analysis, the central bursting defect in plane strain drawing and extrusion is analyzed by comparing the proportional flow with the central bursting flow for the metal with voids at the center. A criterion for the unique conditions that promote this defect has been derived. The metal with voids may flow in the identical manner to that of solid strip with no voids to form a sound flow, deterring central bursting. A solid strip, on the other hand, or a material with voids, may flow in a manner so as to produce central bursting defects. A major conclusion of the study is that, for a range of combinations of inclined angle of the die, reduction, and friction, central bursting is expected whether or not the material originally had any voids. On the other hand, central bursting can be prevented even if the original rod contains small-size voids.

Sign in / Sign up

Export Citation Format

Share Document